6 Practical MATLAB® Applications for Engineers
R.1.6 Since δ(t) is not a conventional signal, it is not possible to generate a function that
has exactly the same properties as δ(t). However, the Dirak function as well as its
derivatives (dδ(t)/dt) can be approximated by different mathematical models.
Some of the approximations are listed as follows (Lathi, 1998):
( ) lim
sin( / )
/(sin)()
t
ata
tact ait→
01
using ’sllim ( )( ) lim/
ititajt aate
jtt
e→→−
0
0using exponentials(^222) 4/
()
a
a
using GaussianR.1.7 Multiplying a unit impulse δ(t) by a constant A changes the area of the impulse to
A, or the amplitude of the impulse becomes A.
R.1.8 The impulse function δ(t) when multiplied by an arbitrary function f(t) results in an
impulse with the magnitude of the function evaluated at t = 0 , indicated by
(t) f(t) = f(0) (t)
Observe that f(0) δ(t) can be defi ned as
ftt
ft()()
()
0
00
00
for
for
R.1.9 A shifted impulse δ(t − t 1 ) is illustrated in Figure 1.3. When the shifted impulse
δ(t − t 1 ) is multiplied by an arbitrary function f(t), the result is given by
(t − t 1 ) ⋅ f(t) = f(t 1 ) ⋅ (t − t 1 )
R.1.10 The derivative of the unit Dirak δ(t) i s ca l led t he u n it doublet, denoted by
d[δ(t)]
dt
= δ′(t),
is illustrated in Figure 1.4.0t 1 tAmplitudeFIGURE 1.
Plot of δ(t − t 1 ).FIGURE 1.
Plot of δ(t)′ as a approaches zero.a1t(t)′